To learn parameters \(\psi \) , we minimize the KL divergence between \(q_{\psi  }(z|y;x)\)  and \(p(z|y;x)\) , which leads to maximizing the evidence lower bound (ELBO) [1]} of the log marginalized likelihood \(log\ p(y|x)\) , as follows (detailed derivation in Supplementary Material):
\(\begin{aligned}&\underset{\psi }{min}\, \, \mathcal {KL}[q_{\psi  }(z|y;x)||p(z|y;x)]\\ &=\underset{\psi }{min}\, \, \mathcal {KL}[q_{\psi  }(z|y;x)||p(z)] -\mathbb {E}_{q}[log\, p(y|z;x)]\\ &+log\ p(y|x).\end{aligned}\) 
